On the Relation Between Complex Modes and Wave Propagation Phenomena (original) (raw)

2002, Journal of Sound and Vibration

This paper discusses the well-known, but often misunderstood, concept of complex modes of dynamic structures. It shows how complex modes can be interpreted in terms of wave propagation phenomena caused by either localized damping or propagation to the surrounding media. Numerical simulation results are presented for di!erent kinds of structures exhibiting modal and wave propagation characteristics: straight beams, an L-shaped beam, and a three-dimensional frame structure. The input/output transfer relations of these structures are obtained using a spectral formulation known as the spectral element method (SEM). With this method, it is straightforward to use in"nite elements, usually known as throw-o! elements, to represent the propagation to in"nity, which is a possible cause of modal complexity. With the SEM model, the exact dynamic behavior of structures can be investigated. The mode complexity of these structures is investigated. It is shown that mode complexity characterizes a behavior that is halfway between purely modal and purely propagative. A coe$cient for quantifying mode complexity is introduced. The mode complexity coe$cient consists of the correlation coe$cient between the real and imaginary parts of the eigenvector, or of the operational de#ection shape (ODS). It is shown that, far from discontinuities, this coe$cient is zero in the case of pure wave propagation in which case the plot of the ODS in the complex plane is a perfect circle. In the other extreme situation, a "nite structure without damping (or with proportional damping), where the mode shape (or the ODS) is a straight line on the complex plane, has a unitary complexity coe$cient. For simple beam structures, it is shown that the mode complexity factor can also be calculated by curve-"tting the mode to an ellipse and computing the ratio of its radii.